In the ornate domed president's room of Woolsey Hall, the governing board of Yale University yesterday introduced the school's new president, Benno C. Schmidt Jr., scion to an academic and institutional legacy begun 284 years ago in New Haven.

Schmidt, 43, is a 1963 graduate of Yale and a 1966 graduate of its law school. He will be the university's 20th president.

With outgoing President Angelo Bartlett Giamatti at his side, Schmidt acknowledged the importance, both real and symbolic, of the role he will assume next July.

"Yale is one of the signal achievements of the experience that is America," he said. "Yale's superb students must be encouraged to engage and sustain original inquiry on the frontiers of knowledge."

"I am humbled before the challenges of this job, before the distinction of Yale's current president and his predecessors, and before the excellence, energy and decency of this unique institution," Schmidt said.

After his introductory remarks, Schmidt was bombarded with questions ranging from his views on divestment from South African-related firms to the impact of the Reagan administration's cuts in aid to education.

The bespectacled lawyer-professor waded cautiously through the torrent of questions, promising to study each in more detail and to "sit down at Bart's feet and learn about this place from the maestro" during their six months of transition.

The intense interest in Schmidt's opinions on controversial topics bespoke the national significance of the job of the Yale president, who is expected to be a leading spokesman on academic and social issues.

Schmidt, dean of Columbia University's law school and a respected author on constitutional law and free speech issues, promised to concentrate his time on scholastic pursuits and not become bogged down in administration.

"I want to be deeply involved in the academic work that goes on at Yale," he said. "I am determined that the administrative aspects of the job should not swamp my academics."

Acknowledging that taking the job "has not been an easy decision," Schmidt told a reporter: "The job is too big for any one person. My own approach is going to be to rely very, very heavily on the wisdom of others."